First-Order DEs — Differential Equations

What Is a Differential Equation?

A DE is an equation involving a function and its derivatives. The order is the highest derivative present. A first-order DE has the form dy/dx = f(x, y).

General form: F(x, y, y') = 0
Solution: a function y = φ(x) satisfying the equation
General solution: family of solutions with arbitrary constant C
Particular solution: satisfies an initial condition y(x₀) = y₀

Separable Equations

Form: dy/dx = g(x)·h(y)
Method: (1/h(y)) dy = g(x) dx → integrate both sides

Example: dy/dx = xy

(1/y) dy = x dx → ln|y| = x²/2 + C → y = Ae^(x²/2)

Uses integration and exponential functions.

First-Order Linear

Form: dy/dx + P(x)y = Q(x)
Integrating factor: μ(x) = e^(∫P(x)dx)
Solution: y = (1/μ) ∫ μ·Q dx

The integrating factor technique converts a non-separable DE into an exact derivative. This method uses the product rule in reverse and relies on integration techniques.

Exact Equations

Form: M(x,y)dx + N(x,y)dy = 0
Exact if: ∂M/∂y = ∂N/∂x
Solution: find F where ∂F/∂x = M and ∂F/∂y = N; then F(x,y) = C

Exactness connects to partial derivatives and conservative vector fields. Non-exact equations can sometimes be made exact with an integrating factor.

Applications

Exponential growth/decay: dy/dt = ky → y = y₀eᵏᵗ (population, radioactive decay)
Newton's cooling: dT/dt = −k(T − Tₐ) → T = Tₐ + (T₀ − Tₐ)e⁻ᵏᵗ
Mixing problems: Rate in − rate out → first-order linear DE
Logistic growth: dP/dt = rP(1 − P/K) → S-shaped curve — uses partial fractions to integrate

First-order DEs appear in every scientific field. In systems of DEs, x' = Ax, eigenvalues determine whether solutions grow, decay, or oscillate. In probability, the exponential distribution's memoryless property comes from the DE: f'(t) = −λf(t).

← Back toDE Overview Next Lesson →Second-Order DEs

First-Order Differential Equations